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Register a new userTwisted Sequences of Extensions
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Authors
Kevin J. Carlin
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Gabber and Joseph introduced a ladder diagram between two natural sequences of extensions. Their diagram is used to produce a twisted sequence that is applied to old and new results on extension groups in category $mathcal{O}$.
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We give some applications of a Hopf algebra constructed from a group acting on another Hopf algebra A as Hopf automorphisms, namely Molnars smash coproduct Hopf algebra. We find connections between the exponent and Frobenius-Schur indicators of a smash coproduct and the twisted exponents and twisted Frobenius-Schur indicators of the original Hopf algebra A. We study the category of modules of the smash coproduct.
For a real bounded symmetric domain, G/K, we construct various natural enlargements to which several aspects of harmonic analysis on G/K and G have extensions. Our starting point is the realization of G/K as a totally real submanifold in a bounded domain G_h/K_h. We describe the boundary orbits and relate them to the boundary orbits of G_h/K_h. We relate the crown and the split-holomorphic crown of G/K to the crown Xi_h of G_h/K_h. We identify an extension of a representation of K to a larger group L_c and use that to extend sections of vector bundles over the Borel compactification of G/K to its closure. Also, we show there is an analytic extension of K-finite matrix coefficients of G to a specific Matsuki cycle space.
A double extension ($mathscr{D}$ extension) of a Lie (super)algebra $mathfrak a$ with a non-degenerate invariant symmetric bilinear form $mathscr{B}$, briefly: a NIS-(super)algebra, is an enlargement of $mathfrak a$ by means of a central extension and a derivation; the affine Kac-Moody algebras are the best known examples of double extensions of loops algebras. Let $mathfrak a$ be a restricted Lie (super)algebra with a NIS $mathscr{B}$. Suppose $mathfrak a$ has a restricted derivation $mathscr{D}$ such that $mathscr{B}$ is $mathscr{D}$-invariant. We show that the double extension of $mathfrak a$ constructed by means of $mathscr{B}$ and $mathscr{D}$ is restricted. We show that, the other way round, any restricted NIS-(super)algebra with non-trivial center can be obtained as a $mathscr{D}$-extension of another restricted NIS-(super)algebra subject to an extra condition on the central element. We give new examples of $mathscr{D}$-extensions of restricted Lie (super)algebras, and pre-Lie superalgebras indigenous to characteristic 3.
Suppose that $E=A[x;sigma,delta]$ is an Ore extension with $sigma$ an automorphism. It is proved that if $A$ is twisted Calabi-Yau of dimension $d$, then $E$ is twisted Calabi-Yau of dimension $d+1$. The relation between their Nakayama automorphisms is also studied. As an application, the Nakayama automorphisms of a class of 5-dimensional Artin-Schelter regular algebras are given explicitly.
A recent result of N. Abe implies that the Gabber-Joseph conjecture is true for the first-degree extensions between Verma modules with regular integral highest weights.
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